Transactions of the American Mathematical Society

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Normal structure of the one-point stabilizer of a doubly-transitive permutation group. II


Author: Michael E. O’Nan
Journal: Trans. Amer. Math. Soc. 214 (1975), 43-74
MSC: Primary 20B20
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Abstract: The main result is that the socle of the point stabilizer of a doubly-transitive permutation group is abelian or the direct product of an abelian group and a simple group. Under certain circumstances, it is proved that the lengths of the orbits of a normal subgroup of the one point stabilizer bound the degree of the group. As a corollary, a fixed nonabelian simple group occurs as a factor of the socle of the one point stabilizer of at most finitely many doubly-transitive groups.


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  • [1] Helmut Bender, Transitive Gruppen gerader Ordnung, in denen jede Involution genau einen Punkt festläßt, J. Algebra 17 (1971), 527–554 (German). MR 0288172
  • [2] Richard Brauer and Michio Suzuki, On finite groups of even order whose 2-Sylow group is a quaternion group, Proc. Nat. Acad. Sci. U.S.A. 45 (1959), 1757–1759. MR 0109846
  • [3] Walter Feit and John G. Thompson, Solvability of groups of odd order, Pacific J. Math. 13 (1963), 775–1029. MR 0166261
  • [4] George Glauberman, Central elements in core-free groups, J. Algebra 4 (1966), 403–420. MR 0202822
  • [5] George Glauberman, On the automorphism groups of a finite group having no non-identity normal subgroups of odd order, Math. Z. 93 (1966), 154–160. MR 0194503
  • [6] Daniel Gorenstein and John H. Walter, On finite groups with dihedral Sylow 2-subgroups, Illinois J. Math. 6 (1962), 553–593. MR 0142619
  • [7] O. Grun, Beitrage zur Gruppen theorie. I, J. Reine Angew. Math. 174 (1935), 1-14.
  • [8] Michael O’Nan, A characterization of 𝐿_{𝑛}(𝑞) as a permutation group, Math. Z. 127 (1972), 301–314. MR 0311748
  • [9] -, Normal structure of the one-point stabilizer of a doubly-transitive permutation group. I. Trans. Amer. Math. Soc. 214 (1975), 43-74
  • [10] B. Shult, On the fusion of an involution in its centralizer (unplublished).
  • [11] Michael Aschbacher, 𝔉-sets and permutation groups, J. Algebra 30 (1974), 400–416. MR 0347952

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-75-99942-0
Keywords: Doubly-transitive, permutation group, one-point stabilizer
Article copyright: © Copyright 1975 American Mathematical Society